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Dissecting a Regular Triangle into Three Similar Tiles

Two figures are called similar if they have the same shape; they may have different sizes.
This Demonstration presents solutions to three closely connected dissection problems, one of them trivial, one of them easy, and one of them difficult. The problem is to cut a regular triangle into three similar pieces.
If the tiles can all be the same size, then there are many solutions (challenge 1).
If the tiles must all be different sizes, then there is one solution (challenge 2).
If two of the three tiles must be the same size, the solution is a polyiamond (challenge 3).
There is also a fractal solution to the mixed case.

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Dissecting a regular triangle into three similar polygons, two of them of the same size, looks quite impossible at first glance.
Hence it is no wonder that this problem was unsolved until 1987, when the author of this Demonstration stumbled upon it as part of working on his book A Puzzling Journey To The Reptiles And Related Animals on irregular rep-tiles. The surprising solution can also be found in his Zilllions game 'Isolattice'. The solution became widely known when Martin Gardner published it.
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